File size: 4,631 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Geometry.IsometricImmersion.Pulses.QPulseFirstJetBudget
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.HighEquation
theorem iteratedCoordPartial_sub_actual {f h : Coord → ℝ} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hh : ContDiffOn ℝ ∞ h U) (hU : IsOpen U)
(ds : List (Fin 2)) {p : Coord} (hp : p ∈ U) :
iteratedCoordPartial ds (fun q => f q-h q) p =
iteratedCoordPartial ds f p-iteratedCoordPartial ds h p := by
rw [iteratedCoordPartial_eq_jet (hf.sub hh) hU ds hp]
change iteratedFDeriv ℝ ds.length (f-h) p (coordinateDirections ds) = _
rw [iteratedFDeriv_sub_apply (i := ds.length)
((hf.contDiffAt (hU.mem_nhds hp)).of_le (WithTop.coe_le_coe.mpr le_top))
((hh.contDiffAt (hU.mem_nhds hp)).of_le (WithTop.coe_le_coe.mpr le_top)),
sub_apply,
←iteratedCoordPartial_eq_jet hf hU ds hp,←iteratedCoordPartial_eq_jet hh hU ds hp]
theorem actual_metric_word_difference_le {g h : MetricField} {U : Set Coord}
(hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U)
{p : Coord} (hp : p ∈ U) (i j : Fin 2) (ds : List (Fin 2)) {epsilon : ℝ}
(hjet : ‖iteratedFDeriv ℝ ds.length (fun q => g q i j-h q i j) p‖ ≤ epsilon) :
|iteratedCoordPartial ds (fun q => g q i j) p-
iteratedCoordPartial ds (fun q => h q i j) p| ≤ epsilon := by
have hb := (norm_iteratedCoordPartial_le_jet ((hg.1 i j).sub (hh.1 i j)) hU ds hp).trans hjet
rw [iteratedCoordPartial_sub_actual (hg.1 i j) (hh.1 i j) hU ds hp,
Real.norm_eq_abs] at hb
exact hb
theorem actual_metric_jet_approximation_inputs {g h : MetricField} {U : Set Coord}
(hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U)
{p : Coord} (hp : p ∈ U) {epsilon : ℝ}
(hjets : ∀ i j : Fin 2, ∀ k ≤ 2,
‖iteratedFDeriv ℝ k (fun q => g q i j-h q i j) p‖ ≤ epsilon) :
‖actualCurvatureFirstInput g p-actualCurvatureFirstInput h p‖ ≤ epsilon ∧
∀ i j k l : Fin 2,
|coordPartial i (coordPartial j (fun q => g q k l)) p-
coordPartial i (coordPartial j (fun q => h q k l)) p| ≤ epsilon := by
have he : 0 ≤ epsilon := (norm_nonneg _).trans (hjets 0 0 0 (by omega))
have hval (i j : Fin 2) : |g p i j-h p i j| ≤ epsilon := by
simpa only [iteratedCoordPartial] using actual_metric_word_difference_le hg hh hU hp i j []
(hjets i j 0 (by omega))
have hder (d i j : Fin 2) :
|coordPartial d (fun q => g q i j) p-coordPartial d (fun q => h q i j) p| ≤ epsilon := by
simpa only [iteratedCoordPartial] using actual_metric_word_difference_le hg hh hU hp i j [d]
(hjets i j 1 (by omega))
constructor
· change max ‖(fun i j => g p i j-h p i j)‖
‖(fun d i j => coordPartial d (fun q => g q i j) p-
coordPartial d (fun q => h q i j) p)‖ ≤ epsilon
apply max_le
· apply (pi_norm_le_iff_of_nonneg he).mpr
intro i
apply (pi_norm_le_iff_of_nonneg he).mpr
intro j
simpa only [Real.norm_eq_abs] using hval i j
· apply (pi_norm_le_iff_of_nonneg he).mpr
intro d
apply (pi_norm_le_iff_of_nonneg he).mpr
intro i
apply (pi_norm_le_iff_of_nonneg he).mpr
intro j
simpa only [Real.norm_eq_abs] using hder d i j
· intro i j k l
simpa only [iteratedCoordPartial] using actual_metric_word_difference_le hg hh hU hp k l [i,j]
(hjets k l 2 (by omega))
theorem actual_sheared_first_jet_error_from_Ctau
{gStar gTau : MetricField} {U : Set Coord} {q0 a delta : ℝ} {N n : ℕ}
(hgStar : SmoothPositiveOn gStar U)
(hgTest : SmoothPositiveOn (testMetric gStar q0 a N delta n) U)
(hgTau : SmoothPositiveOn gTau U) (hU : IsOpen U)
(hq : |q0| ≤ 1) (ha : 0 < a) (hN : 0 < N) (hn : 2 ≤ n) (hlarge : N-1 ≤ n)
{p : Coord} (hp : inverseShearCoordinates q0 p ∈ U)
(hjets : ∀ i j : Fin 2, ∀ k ≤ n,
‖iteratedFDeriv ℝ k (fun q => gTau q i j-testMetric gStar q0 a N delta n q i j)
(inverseShearCoordinates q0 p)‖ ≤ 1/(n : ℝ)^n) :
‖actualCurvatureFirstInput (metricInShearCoordinates gTau q0) p-
actualCurvatureFirstInput (metricInShearCoordinates gStar q0) p‖ ≤
qPulseFirstJetBudget a ha N delta n := by
apply actual_sheared_first_jet_error_nat hgStar hgTest hgTau hU hq ha hN
(by omega) hlarge hp
exact (actual_metric_jet_approximation_inputs hgTau hgTest hU hp
(fun i j k hk => hjets i j k (hk.trans hn))).1
end SmoothLocal.Pulse
end
end OAI
|